Directory UMM :Journals:Journal_of_mathematics:OTHER:

185

Documenta Math.

Rolling Factors Deformations
and Extensions of Canonical Curves

Jan Stevens

Received: October 10, 2000
Communicated by Thomas Peternell

Abstract. A tetragonal canonical curve is the complete intersection
of two divisors on a scroll. The equations can be written in ‘rolling
factors’ format. For such homogeneous ideals we give methods to
compute infinitesimal deformations. Deformations can be obstructed.
For the case of quadratic equations on the scroll we derive explicit base
equations. They are used to study extensions of tetragonal curves.
2000 Mathematics Subject Classification: 14B07 14H51 14J28 32S30
Keywords and Phrases: Tetragonal curves, rolling factors, K3 surfaces
An easy dimension count shows that not all canonical curves are hyperplane

sections of K3 surfaces. A surface with a given curve as hyperplane section is
called an extension of the curve. With this terminology, the general canonical
curve has only trivial extensions, obtained by taking a cone over the curve. In
this paper we concentrate on extensions of tetragonal curves.
The extension problem is related to deformation theory for cones. This is best
seen in terms of equations. Suppose we have coordinates (x0 : · · · : xn : t) on
Pn+1 with the special hyperplane section given by t = 0. We describe an
extension W of a variety V : fj (xi ) = 0 by a system of equations Fj (xi , t) = 0
with Fj (xi , 0) = fj (xi ). We write Fj (xi , t) = fj (xi ) + tfj′ (xi ) + · · · + aj tdj ,
where dj is the degree of Fj . Considering (x0 , . . . , xn , t) as affine coordinates
on Cn+1 × C we can read the equations in a different way. The equations
fj (xi ) = 0 define the affine cone C(V ) over V and Fj (xi , t) = 0 describes a 1parameter deformation of C(V ). The corresponding infinitesimal deformation
is fj (xi ) 7→ fj′ (xi ), which is a deformation of weight −1. Conversely, given a
1-parameter deformation Fj (xi , t) = 0 of C(V ), with Fj homogeneous of degree
dj , we get an extension W of V . For most of the cones considered here the
only infinitesimal deformations of negative weight have weight −1 and in that
case the versal deformation in negative weight gives a good description of all
possible extensions.
Documenta Mathematica 7 (2002) 185–226

186

Jan Stevens

As the number of equations typically is much larger than the codimension one
needs good ways to describe them. A prime example is a determinantal scheme
X: its ideal is generated by the t × t minors of an r × s matrix, which gives
a compact description of the equations. Following Miles Reid we call this a
format. Canonical curves are not themselves determinantal, but they do lie on
scrolls: a k-gonal curve lies on a (k − 1)-dimensional scroll, which is given by
the minors of a 2 × (g − k + 1) matrix. For k = 3 the curve is a divisor on the
scroll, given by one bihomogeneous equation, and for k = 4 it is a complete
intersection, given by two bihomogeneous equations. In these cases there is a
simple procedure (‘rolling factors’) to write out one resp. two sets of equations
on Pg−1 cutting out the curve on the scroll.
Powerful methods exist to compute infinitesimal deformations without using
explicit equations. We used them for the extension problem for hyperelliptic
curves of high degree [Stevens 1996] and trigonal canonical curves [Drewes–
Stevens 1996]. In these papers also several direct computations with the equations occur. They seem unavoidable for tetragonal curves, the subject of a
preprint by James N. Brawner [Brawner 1996]. The results of these computations do not depend on the particular way of choosing the equations cutting

out the curve on the scroll. This observation was the starting point of this
paper.
We distinguish between different types of deformations and extensions. If only
the equations on the scroll are deformed, but not the scroll itself we speak of
pure rolling factors deformations. A typical extension lies then on the projective cone over the scroll. Such a cone is a special case of a scroll of one
dimension higher. If the extension lies on a scroll which is not a cone, the
equations of the scroll are also deformed. We have a rolling factors deformation. Finally if the extension does not lie on a scroll of one dimension higher
we are in the situation of a non-scrollar deformation. Non-scrollar extensions
of tetragonal curves occur only in connection with Del Pezzo surfaces. Not every infinitesimal deformation of a scroll gives rise to a deformation of complete
intersections on it. One needs certain lifting conditions, which are linear equations in the deformation variables of the scroll. Our first main result describes
them, depending only on the coefficients of the equations on the scroll.
The next problem is to extend the infinitesimal deformations to a versal deformation. Here we restrict ourselves to the case that all defining equations
are quadratic. Our methods thus do not apply to trigonal curves, but we can
handle tetragonal curves. Rolling factors obstructions arise. Previously we observed that one can write them down, given explicit equations on Pn [Stevens
1996, Prop. 2.12]. Here we give formulas depending only on the coefficients of
the equations on the scroll. As first application we study base spaces for hyperelliptic cones. The equations have enough structure so that explicit solutions
can be given.
Surfaces with canonical hyperplane sections are a classical subject. References
to the older literature can be found in Epema’s thesis [Epema 1983], which is
especially relevant for our purposes. His results say that apart from K3 surfaces

Documenta Mathematica 7 (2002) 185–226

Rolling Factors Deformations

187

only rational surfaces or birationally ruled surfaces can occur. Furthermore he
describes a construction of such surfaces. Extensions of pure rolling factors type
of tetragonal curves fit very well in this description. A general rolling factors
extension is a complete intersection on a nonsingular four-dimensional scroll.
The classification of such surfaces [Brawner 1997], which we recall below, shows
that surfaces with isolated singularities and in particular K3s can only occur
if the degrees of the equations on the scroll differ at most by 4. A tetragonal
curve of high genus with general discrete invariants has no pure rolling factors
deformations. Extensions exist if the base equations have a solution. For low
genus we have more variables than equations. For the maximal genus where
almost all curves have a K3 extension we find:
Proposition. The general tetragonal curve of genus 15 is hyperplane section
of 256 different K3 surfaces.
We also look at examples with genus 16 and 17. It is unclear to us which

property of a curve makes it have an extension (apart from the property of
being a hyperplane section).
The contents of this paper is as follows. First we describe the rolling factors
format and explain in detail the equations and relations for the complete intersection of two divisors on a scroll. Next we recall how canonical curves fit into
this pattern. In particular we describe the discrete invariants for tetragonal
curves. The same is done for K3 surfaces. The second section is devoted to
the computation of infinitesimal deformations. First non-scrollar deformations
are treated, followed by rolling factors deformations. The main result here describes the lifting matrix. As application the dimension of T 1 is determined for
tetragonal cones. In the third section the base equations for complete intersections of quadrics on scrolls are derived. As examples base spaces for hyperelliptic cones are studied. The final section describes extensions of tetragonal
curves.
1. Rolling factors format.
A subvariety of a determinantal variety can be described by the determinantal
equations and additional equations obtained by ‘rolling factors’ [Reid 1989]. A
typical example is the case of divisors on scrolls.
We start with a k-dimensional rational normal scroll S ⊂ Pn (for the theory
of scrolls we refer to [Reid 1997]). The classical construction is to take k complementary linear subspaces Li spanning Pn , each containing a parametrized
rational normal curve φi : P1 → Ci ⊂ Li of degree di = dim Li ,P
and to take for
each p ∈ P1 the span of the points φi (p). The degree of S is d =
di = n−k+1.

If all di > 0 the scroll S is a Pd−1 -bundle over P1 . We allow however that di = 0
for some i. Then S is the image of Pd−1 -bundle Se over P1 and Se → S is a
rational resolution of singularities.
To give a coordinate description, we take homogeneous coordinates (s : t) on
P1 , and (z (1) : · · · : z (k) ) on the fibres. Coordinates on Pn are zj(i) = z (i) sdi −j tj ,
Documenta Mathematica 7 (2002) 185–226

188

Jan Stevens

with 0 ≤ j ≤ di , 1 ≤ i ≤ k. We give the variable z (i) the weight −di . The
scroll S is given by the minors of the matrix
Φ=

µ

z0(1)
z1(1)

...
...

zd(1)
1 −1
zd(1)
1

...
...

z0(k)
z1(k)

...
...

zd(k)
k −1
zd(k)

k

¶

.

We now consider a divisor on Se in the linear system |aH − bR|, where the
e When we
hyperplane class H and the ruling R generate the Picard group of S.
speak of degree on Se this will be with respect to H. The divisor can be given
by one bihomogeneous equation P (s, t, z (i) ) of degree a in the z (i) , and total
degree −b. By multiplying P (s, t, z (i) ) with a polynomial of degree b in (s:t) we
obtain an equation of degree 0, which can be expressed as polynomial of degree
a in the zj(i) ; this expression is not unique, but the difference of two expressions
lies in the ideal of the scroll. By the obvious choice, multiplying with sb−m tm ,
we obtain b + 1 equations Pm . In the transition from the equation Pm to Pm+1
we have to increase by one the sum of the lower indices of the factors zj(i) in
each monomial, and we can and will always achieve this by increasing exactly
one index. This amounts to replacing a zj(i) , which occurs in the top row of the
(i)

in the bottom row of the same column. This is
matrix, by the element zj+1
the procedure of ‘rolling factors’.
Example 1.1. Consider the cone over 2d − b points in Pd , lying on a rational
normal curve of degree d, with b < d. Let the polynomial P (s, t) = p0 s2d−b +
p1 s2d−b−1 t + · · · + p2d−b t2d−b determine the points on the rational curve. We
get the determinantal
¯
¯
¯ z0 z1 . . . zd−1 ¯
¯
¯
¯ z1 z2 . . .
zd ¯

and additional equations Pm . To be specific we assume that b = 2c:
P0
P1
P2c

2
=
p0 z02 + p1 z0 z1 + · · · + p2d−2c−1 zd−c−1 zd−c + p2d−2c zd−c
2
2
=
p0 z0 z1 + p1 z1 + · · · + p2d−2c−1 zd−c + p2d−2c zd−c zd−c+1
..
.
= p0 zc2 + p1 zc zc+1 + · · · + p2d−2c−1 zd−1 zd + p2d−2c zd2 .

The ‘rolling factors’ phenomenon can also occur if the entries of the matrix are
more general.
Example 1.2. Consider a non-singular hyperelliptic curve of genus 5, with
are Weierstrass
a half-canonical line bundle L = g12 + P1 + P2 where the Pi L
points. According to [Reid 1989], Thm. 3, the ring R(C, L) =
H 0 (C, nL) is
k[x1 , x2 , y1 , y2 , z1 , z2 ]/I with I given by the determinantal
¯

¯ x1
¯
¯ x2

y1
x21

x22
y2

¯
z1 ¯¯
z2 ¯

Documenta Mathematica 7 (2002) 185–226

Rolling Factors Deformations

189

and the three rolling factors equations
z12 = x21 h + y13 + x42 y2
z1 z2 = x1 x2 h + y12 x21 + x22 y22
z22 = x22 h + y1 x41 + y23
where h is some quartic in x1 , x2 , y1 , y2 .
The description of the syzygies of a subvariety V of the scroll S proceeds in two
steps. First one constructs a resolution of OVe by vector bundles on Se which
are repeated extensions of line bundles. Schreyer describes, following Eisenbud,
Eagon-Northcott type complexes C b such that C b (a) is the minimal resolution of
¡
¢
i∗ OSe(−aH + bR) as OPn -module, if b ≥ −1 [Schreyer 1986]. Here i: Se → Pn
is the map defined by H. The resolution of OV is then obtained by taking an
(iterated) mapping cone.
The matrix Φ defining the scroll can be obtained intrinsically from the multiplication map
H 0 OSe(R) ⊗ H 0 OSe(H − R) −→ H 0 OSe(H) .

In general, given a map Φ: F → G of locally free sheaves of rank f and g
respectively, f ≥ g, on a variety one defines Eagon-Northcott type complexes
C b , b ≥ −1, in the following way:
Cjb

½ Vj
F ⊗ S G,
for 0 ≤ j ≤ b
= Vj+g−1 b−j
Vg ∗
F ⊗ Dj−b−1 G∗ ⊗
G , for j ≥ b + 1

with
by multiplication with Φ ∈ F ∗ ⊗ G for j 6= b + 1 and
Vg differential
Vg ∗ defined
Vg
Φ∈
F ⊗
G for j = b + 1 in the appropriate term of the exterior,
symmetric or divided power algebra.
In our situation F ∼
= OP2n with Φ given by the matrix of the
= OPdn (−1) and G ∼
b
scroll. Then C (−a) is for b ≥ −1 the minimal resolution of OSe(−aH + bR) as
OP -module [Schreyer 1986, Cor. 1.2].
Now let V ⊂ S ⊂ Pn be a ‘complete intersection’ of divisors Yi ∼ ai H −
bi R, i = 1, . . . , l, on a k-dimensional rational scroll of degree d with bi ≥ 0.
The resolution of OV as OS -module is a Koszul complex and the iterated
mapping cone of complexes C b is the minimal resolution [Schreyer 1986, Sect. 3,
Example].
To make this resolution more explicit we look at the case l = 2, which is relevant
for tetragonal curves. The iterated mapping cone is
¤
£ b1 +b2
(−a1 − a2 ) −→ C b1 (−a1 ) ⊕ C b2 (−a2 ) −→ C 0
C
To describe equations and relations we give the first steps of this complex. We
first consider the case that b1 ≥ b2 > 0. We write O for OPn . We get the
Documenta Mathematica 7 (2002) 185–226

190

Jan Stevens

double complex
O

←−

x



V2

O d (−1)

x



←−

V3

O d (−1)⊗O 2

Sb1 O 2 (−a1 )⊕Sb2 O 2 (−a2 ) ←− O d (−1)⊗Sb1 −1 O 2 (−a1 )⊕O d (−1)⊗Sb2 −1 O 2 (−a2 )

x



Sb1 +b2 O 2 (−a1 −a2 )

The equations for V consist of the determinantal ones plus two sets of additional
equations obtained by rolling factors: the two equations P (1) , P (2) defining V
(2)
(1)
.
and b2 + 1 equations Pm
on the scroll give rise to b1 + 1 equations Pm
To describe the relations we introduce the following notation. A column in the
(i)
). We write symbolically (zα , zα+1 ), where the
matrix Φ has the form (zj(i) , zj+1
and
α + 1 means adding 1 to the lower index.
index α stands for the pair (i)
j
′)
(i)
′
′
′
More generally, if α = j and α = (i
j ′ then the sum α + α := j + j only
involves the lower indices. To access the upper index we say that α is of type
i. The rolling factors assumption is that two consecutive additional equations
are of the form
X
Pm =
pα,m zα ,
α

Pm+1 =

X

pα,m zα+1 .

α

where the polynomials pα,m depend on the z-variables and the sum runs over
all possible pairs α = (i)
j . To roll from Pm+1 to Pm+2 we collect the
P ‘coefficients’
in the equation Pm+1 in a different way: we also have Pm+1 = α pα,m+1 zα .
We write the scrollar equations as fαβ = zα zβ+1 − zα+1 zβ . The relations
between them are
Rα,β,γ = fα,β zα − fα,γ zβ + fβ,γ zα ,
Sα,β,γ = fα,β zα+1 − fα,γ zβ+1 + fβ,γ zα+1 ,
V3 d
which corresponds to the term
O (−1) ⊗ O 2 in Schreyer’s resolution. The
(n)
:
second line yields relations involving the two sets of Pm
X
(n)
n
(n)
Rβ,m
= Pm+1
zβ − P m
zβ+1 −
fβ,α p(n)
α,m ,
α

where n = 1, 2 and 0 ≤ m < bi . We note the following relation:
X
n
n
n
(n)
(n)
zγ − Rβ,m
zβ −
Rβ,γ,α
p(n)
Rβ,m
α,m = Pm fβ,γ − fβ,γ Pm .
The right hand side is a Koszul relation; the second factor in each product
is considered as coefficient. There are similar expressions involving zγ+1 , zβ+1
Documenta Mathematica 7 (2002) 185–226

Rolling Factors Deformations

191

and Sβ,γ,α . Finally by multiplication with suitable powers of s and t the Koszul
relation P (1) P (2) − P (2) P (1) gives rise to b1 + b2 + 1 relations — this is the term
Sb1 +b2 O2 (−a1 − a2 ).
In case b1 > b2 = 0 the resolution is
V2 d
V3 d
O
←
O (−1) ←
O (−1) ⊗ O 2
x
x




V2 d
O (−1 − a2 )
Sb1 O2 (−a1 ) ⊕ O2 (−a2 ) ← Od (−1) ⊗ Sb1 −1 O2 (−a1 ) ⊕
x


Sb1 O2 (−a1 − a2 )

The new term expresses the Koszul relations between the one equation P (2) and
the determinantal equations (which had previously been expressible in terms of
rolling factors relations). For the computation of deformations these relations
may be ignored.
Finally, if b2 = −1, the equations change drastically.
(1.3) Canonical curves [Schreyer 1986].
A k-gonal canonical curve lies on a (k − 1)-dimensional scroll of degree d = g −
k + 1. We write D for the divisor of the gk1 . To describe the type S(e1 , ..., ek−1 )
of the scroll we introduce the numbers
fi = h0 (C, K − iD) − h0 (C, K − (i + i)D) = k + h0 (iD) − h0 ((i + 1)D)
for i ≥ 0 and set
ei = #{j | fj ≥ i} − 1 .
In particular, e1 is the minimal number i such that h0 ((i + 1)D) − h0 (iD) = k
and it satisfies therefore e1 ≤ 2g−2
k .
A trigonal curve lies on a scroll of type S(e1 , e2 ) and degree d = e1 + e2 = g − 2
with
2g − 2
g−4
≥ e1 ≥ e2 ≥
3
3
as a divisor of type 3H − (g − 4)R. The minimal resolution of OC is given by
the mapping cone
C d−2 (−3) −→ C 0 .
Introducing bihomogeneous coordinates (x : y; s : t) and coordinates xi =
xse1 −i ti , yi = yse2 −i ti we obtain the scroll
¶
µ
x0 x1 . . . xe1 −1 y0 y1 . . . ye2 −1
y1 y2 . . .
ye2
x1 x2 . . .
xe1
and a bihomogeneous equation for C
P = A2e1 −e2 +2 x3 + Be1 +2 x2 y + Ce2 +2 xy 2 + D2e2 −e1 +2 y 3
Documenta Mathematica 7 (2002) 185–226

192

Jan Stevens

where A2e1 −e2 +2 is a polynomial in (s : t) of degree 2e1 − e2 + 2 and similarly for
the other coefficients. By rolling factors P gives rise to g − 3 extra equations.
can also be explained from the condition that the
The inequality e1 ≤ 2g−2
3
curve C is nonsingular, which implies that the polynomial P is irreducible, and
therefore the degree 2e2 − e1 + 2 = 2g − 2 − 3e1 of the polynomial D2e2 −e1 +2 is
nonnegative. The other inequality follows from this one because e1 = g −e2 −2,
but also by considering the degree of A2e1 −e2 +2 .
A tetragonal curve of genus g ≥ 5 is a complete intersection of divisors Y ∼
2H − b1 R and Z ∼ 2H − b2 R on a scroll of type S(e1 , e2 , e3 ) of degree d =
e1 + e2 + e3 = g − 3, with b1 + b2 = d − 2, and
g−1
≥ e1 ≥ e2 ≥ e3 ≥ 0
2
We introduce bihomogeneous coordinates (x: y : z; s : t). Then Y is given by an
equation
P = P1,1 x2 + P1,2 xy + · · · + P3,3 z 2
with Pij (if nonzero) a polynomial in (s : t) of degree ei + ej − b1 and likewise
Z has equation
Q = Q1,1 x2 + Q1,2 xy + · · · + Q3,3 z 2
with deg Qij = ei + ej − b2 .
The minimal resolution is of type discussed above, because the condition −1 ≤
b2 ≤ b1 ≤ d−1 is satisfied: the only possibility to have a divisor of type 2H −bR
with b ≥ d is to have e1 = e2 = d/2, e3 = 0 and b = d, but then the equation
P is of the form αx2 + βxy + γy 2 with constant coefficients, so reducible. If
b2 = −1 also cubics are needed to generate the ideal, so the curve admits also
a g31 or g52 ; this happens only up to g = 6. We exclude these cases and assume
that b2 ≥ 0.
Lemma 1.4. We have b1 ≤ 2e2 and b2 ≤ 2e3 .
Proof . If b1 > 2e2 the polynomials P22 , P23 and P33 vanish so P is reducible
and therefore C. If b2 > 2e3 then P33 and Q33 vanish. This means that the
section x = y = 0 is a component of Y ∩ Z on the P2 -bundle whose image in
Pg−1 is the scroll (if e3 > 0 the scroll is nonsingular, but for e3 = 0 it is a cone).
As the arithmetic genus of Y ∩ Z is g and its image has to be the nonsingular
curve C of genus g, the line cannot be a component.
¤
This Lemma is parts 2 – 4 in [Brawner 1997, Prop. 3.1]. Its last part is incorrect.
It states that b1 ≤ e1 + e3 if e3 > 0, and builds upon the fact that Y has only
isolated singularities. However the discussion in [Schreyer 1986] makes clear
that this need not be the case.
The surface Y fibres over P1 . There are now two cases, first that the general
fibre is a non-singular conic. In this case one of the coefficients P13 , P23 or P33
is nonzero, giving indeed b1 ≤ e1 + e3 .
Documenta Mathematica 7 (2002) 185–226

Rolling Factors Deformations

193

The other possibility is that each fibre is a singular conic. Then Y is a birationally ruled surface over a (hyper)elliptic curve E with a rational curve
E of double points, the canonical image of E, and C does not intersect E.
This means that the section E of the scroll does not intersect the surface Z,
so if one inserts the parametrisation of E in the equation of Z one obtains a
non-zero constant. Let the section be given by polynomials in (s : t), which if
nonzero have degree ds − e1 , ds − e2 , ds − e3 . Inserting them in the polynomial
Q gives a polynomial of degree 2ds − b2 . So b2 is even and 2ds = b2 ≤ 2e3 .
On the other hand ds − e3 ≥ 0 so ds = e3 and b2 = 2e3 . The genus of E
satisfies pa (E) = b2 /2 + 1. If b1 > e1 + e3 , then Y is singular along the section
x = y = 0. An hyperelliptic involution can also occur if b1 ≤ e1 + e3 .
We have shown:
Lemma 1.5. If Y is singular, in particular if b1 > e1 + e3 , then b2 = 2e3 .
Finally we analyze the case b2 = 0 (cf. [Brawner 1996]).
Lemma 1.6. A nonsingular tetragonal curve is bi-elliptic or lies on a Del Pezzo
surface if and only if b2 = 0. The first case occurs for e3 = 0, and the second for
the values (2, 0, 0), (1, 1, 0), (2, 1, 0), (1, 1, 1), (3, 1, 0), (2, 2, 0), (2, 1, 1), (3, 2, 0),
(2, 2, 1), (4, 2, 0), (3, 2, 1) or (2, 2, 2) of the triple (e1 , e2 , e3 ).
Proof . If the curve is bielliptic or lies on a Del Pezzo, the g41 is not unique,
which implies that the scroll is not unique. This is only possible if b2 = 0 by
[Schreyer 1986], p. 127. Then C is the complete intersection of a quadric and
a surface Y of degree g − 1, which is uniquely determined by C.
The inequality e1 + e2 + e3 − 2 = b1 ≤ 2e2 shows that e3 ≤ e2 − e − 1 + 2 ≤ 2.
If the general fibre of Y over P1 is non-singular we have b1 ≤ e1 + e3 . This
gives e2 ≤ 2 and b1 ≤ 4. The possible values are now easily determined. If
the general fibre of Y is singular then e3 = b2 /2 = 0 and Y is an elliptic cone.
¤
(1.7) K3 surfaces.
Let X be a K3 surface (with at most rational double point singularities) on
a scroll. If the scroll is nonsingular the projection onto P1 gives an elliptic
fibration on X, whose general fibre is smooth. This is even true if the scroll is
e on Se has only isolated singularities.
singular: the strict transform X
We start with the case of divisors. A treatment of such scrollar surfaces with
an elliptic fibration can be found in [Reid 1997, 2.11]. One finds:
Lemma 1.8. For the general F ∈ |3H − kR| on a scroll S(e1 , e2 , e3 ) the general
fibre of the elliptic fibration is a nonsingular cubic curve if and only if k ≤ 3e2
and k ≤ e1 + 2e3 .
If one fixes k and e1 + e2 + e3 these conditions limit the possible distribution of
the integers (e1 , e2 , e3 ). By the adjunction formula one has k = e1 +e2 +e3 −2 for
a K3 surface. In this case we obtain 12 solutions, which fall into 3 deformation
Documenta Mathematica 7 (2002) 185–226

194

Jan Stevens

types of scrolls, according to

P

ei

(mod 3):

(e + 2, e, e − 2) → (e + 1, e, e − 1) → (e, e, e)
(e + 3, e, e − 2) → (e + 2, e, e − 1) → (e + 1, e + 1, e − 1) → (e + 1, e, e)
(e+4, e, e−2) → (e+3, e, e−1) → (e+2, e+1, e−1) → (e+2, e, e) → (e+1, e+1, e)
The general element of the linear system can only have singularities at the base
locus. The base locus is the section (0 : 0 : 1) if and only if k > 3e3 and there is a
singularity at the points (s:t) where both Ae1 +2e3 −k and Ae2 +2e3 −k vanish. The
assumption that the coefficients are general implies now that deg Ae2 +2e3 −k < 0
and deg Ae1 +2e3 −k > 0.
In the 12 cases above this occurs only for (e + 3, e, e − 1) and (e + 2, e, e − 1).
In the first case the term y 2 z is also missing, yielding that there is an A2 singularity at the only zero of Ae2 +2e3 −k , whereas the second case gives an A1 .
The scroll Se+4,e,e−2 deforms into Se+3,e,e−1 , but the general K3-surface on it
does not deform to a K3 on Se+3,e,e−1 , but only those with an A2 -singularity.
These results hold if all ei > 0; we leave the modifications in case e3 = 0 to the
reader.
The tetragonal case is given as exercise in [Reid 1997] and the complete solution
(modulo some minor mistakes) can be found in [Brawner 1997]. We give the
results:
Lemma 1.9. For the general complete intersection of divisors of type 2H −b1 R
and 2H − b2 R on a scroll Se1 ,e2 ,e3 ,e4 the general fibre of the elliptic fibration is
a nonsingular quartic curve if and only if either
α: b1 ≤ e1 + e3 , b1 ≤ 2e2 and b2 ≤ 2e4 , or
β: b1 ≤ e1 + e4 , b1 ≤ 2e2 , 2e4 < b2 ≤ 2e3 and b2 ≤ e2 + e4 .
Proposition 1.10. The general element is singular at a point of the section
(0 : 0 : 0 : 1) if the invariants satisfy in addition one of the following conditions:
1α: b2 < 2e4 , b1 > e1 + e4 .
1βi: b2 ≤ e3 + e4 , e2 + e4 < b1 < e1 + e4 .
1βii: b2 > e3 + e4 , and e1 + e2 + 2e4 > b1 + b2 .
There is a singularity with z 6= 0 if
2α: b1 > e2 + e3 , e1 + e3 > b1 > e1 + e4 .
2αβ: e1 + e4 ≥ b1 > e2 + e3 and
i: if b2 ≤ 2e4 then 2(e1 + e3 + e4 ) > 2b1 + b2
ii: if 2e4 < b2 ≤ e3 + e4 then e1 + 2e3 + e4 > b1 + b2
iii: e3 + e4 < b2 < 2e3
For K3 surfaces we need b1 + b2 = e1 + e2 + e3 + e4 − 2. We give a table listing
the possibilities under this assumption, cf. [Brawner 1997, Table A.1–A.4].
The table lists the possible values for (b1 , b2 ) and gives for each pair the invariants (e1 , e2 , e3 , e4 ) of the scrolls on which the curve can lie. These form
one deformation type with adjacencies going vertically, except Se+2,e,e,e and
Documenta Mathematica 7 (2002) 185–226

195

Rolling Factors Deformations
(b1 , b2 )

(e1 , e2 , e3 , e4 )

M

base

sings

(2e, 2e − 2)

(e + 3, e + 1, e − 1, e − 3)
(e + 3, e, e − 1, e − 2)
(e + 2, e + 1, e − 1, e − 2)
(e + 2, e, e, e − 2)
(e + 2, e, e − 1, e − 1)
(e + 1, e + 1, e − 1, e − 1)
(e + 1, e, e, e − 1)
(e, e, e, e)
(e + 1, e + 1, e, e − 2)
(e + 1, e, e, e − 1)
(e, e, e, e)

17
15
16
16
15
16
17
17
17
17
18

Be−1
Be−1
Be−1
Be−2
Be−1
Be−1
Be−1
∅
Be−2
Be−1
∅

−−
A3
A1
−−
2A1
−−
−−
−−
−−
−−
−−

(e + 4, e + 1, e − 1, e − 3)
(e + 3, e + 1, e − 1, e − 2)
(e + 2, e + 1, e − 1, e − 1)
(e + 1, e + 1, e, e − 1)
(e + 2, e + 1, e, e − 2)
(e + 2, e, e, e − 1)
(e + 1, e + 1, e, e − 1)
(e + 1, e, e, e)

17
16
16
17
17
15
17
18

Be−1
Be−1
Be−1
Be
Be−2
Be−1
Be−1
∅

−−
A1
−−
−−
−−
A1
−−
−−

(e + 5, e + 1, e − 1, e − 3)
(e + 4, e + 1, e − 1, e − 2)
(e + 3, e + 1, e − 1, e − 1)
(e + 2, e + 1, e, e − 1)
(e + 1, e + 1, e + 1, e − 1)
(e + 3, e + 1, e, e − 2)
(e + 2, e + 1, e, e − 1)
(e + 1, e + 1, e, e)
(e + 2, e + 2, e, e − 2)
(e + 2, e + 1, e, e − 1)
(e + 1, e + 1, e + 1, e − 1)
(e + 2, e, e, e)
(e + 1, e + 1, e, e)

18
17
17
18
18
16
16
17
17
16
17
15
17

Be−1
Be−1
Be−1
Be
Be−1
Be
Be
Be
Be−2
Be−1
Be−1
∅
∅

−−
A1
−−
−−
−−
−−
−−
−−
−−
A1
−−
−−
−−

(e + 4, e + 1, e, e − 2)
(e + 3, e + 1, e, e − 1)
(e + 2, e + 1, e, e)
(e + 1, e + 1, e + 1, e)
(e + 3, e + 2, e, e − 2)
(e + 3, e + 1, e, e − 1)
(e + 2, e + 2, e, e − 1)
(e + 2, e + 1, e + 1, e − 1)
(e + 2, e + 1, e, e)
(e + 1, e + 1, e + 1, e)

16
16
17
17
17
15
16
17
16
18

Be
Be
Be
Be
Be
Be
Be
Be−1
Be
Be

A1
A1
A1
A1
−−
A2
A1
−−
−−
−−

(2e − 1, 2e − 1)

(2e + 1, 2e − 2)

(2e, 2e − 1)

(2e + 2, 2e − 2)

(2e + 1, 2e − 1)

(2e, 2e)

(2e + 2, 2e − 1)

(2e + 1, 2e)

Documenta Mathematica 7 (2002) 185–226

196

Jan Stevens

Se+1,e+1,e+1,e−1 which do not deform into each other but are both deformations of Se+2,e+1,e,e−1 and both deform to Se+1,e+1,e,e . Furthermore we give
the number of moduli for each family, in the column M.
In the table we also list the base locus of |2H − b1 R| (which contains that
of |2H − b2 R|). The base locus is a subscroll, for which we use the following
notation [Reid 1997, 2.8]: we denote by Ba the subscroll corresponding to the
subset of all ei with ei ≤ a, defined by the equations z (j) = 0 for ej > a.
We give the number and type of the singularities of the general element; the
number given in the second half of [Brawner 1997, Table A.2] is not correct.
As example of the computations we look at (e + 3, e, e − 1, e − 2) with (b1 , b2 ) =
(2e, 2e − 2). The two equations have the form
p1 xw + p2 xz + p0 y 2 + p3 yx + p6 x2
q0 z + q0 yw + q3 xw + q1 yz + q4 xz + q2 y 2 + q5 yx + q8 x2 ,
2

where the index denotes the degree in (s : t). We first use coordinate transformations to simplify these equations. By replacing y, z and w by suitable
multiples we may assume that the three constant polynomials are 1. Now replacing z by z − 21 q1 y − 12 q4 x removes the yz and xz terms. We then replace
y by y − q3 x to get rid of the xw term. By changing w we finally achieve the
form z 2 + yw + q8 x2 . By a change in (s : t) we may assume that p1 = s. We
now look at the affine chart (w = 1, t = 1) and find y = −z 2 − q8 x2 , which we
insert in the other equation to get an equation of the form x(s + p2 z + · · ·) + z 4 ,
which is an A3 .
We leave it again to the reader to analyze which further singularities can occur
if e4 = 0.
2. Infinitesimal deformations.
Deformations of cones over complete intersections on scrolls need not preserve
the rolling factors format. We shall study in detail those who do. Many
deformations of negative weight are of this type.
Definition 2.1. A pure rolling factors deformation is a deformation in which
the scroll is undeformed and only the equations on the scroll are perturbed.
This means that the deformation of the additional equations can be written
with the rolling factors. Such deformations are always unobstructed. However
this is not the only type of deformation for which the scroll is not changed.
In weight zero one can have deformations inside the scroll, where the type
(b1 , . . . , bl ) changes.
Definition 2.2. A (general) rolling factors deformation is a deformation in
which the scroll is deformed and the additional equations are written in rolling
factors with respect to the deformed scroll.
The equations for the total space of a 1-parameter rolling factors deformation describe a scroll of one dimension higher, containing a subvariety of the
Documenta Mathematica 7 (2002) 185–226

Rolling Factors Deformations

197

same codimension, again in rolling factors format. Deformations over higher
dimensional base spaces may be obstructed. Again in weight zero one can have
deformations of the scroll, where also the type (b1 , . . . , bl ) changes.
Finally there are non-scrollar deformations, where the perturbation of the scrollar equations does not define a deformation of the scroll. Examples of this
phenomenon are easy to find (but difficult to describe explicitly). A trigonal
canonical curve is a divisor in a scroll, whereas the general canonical curve of
the same genus g is not of this type: the codimension of the trigonal locus in
moduli space is g − 4.
Example 2.3. To give an example of a deformation inside a scroll, we let C be
a tetragonal curve in P9 with invariants (2, 2, 2; 3, 1). Then there is a weight 0
deformation to a curve of type (2, 2, 2; 2, 2). To be specific, let C be given by
P = sx2 + ty 2 + (s + t)z 2 , Q = t3 x2 + s3 y 2 + (s3 − t3 )z 2 . We do not deform
the scroll, but only the additional equations:
x20 + y0 y1 + z02 + ε(z12 − x21 )
x0 x1 + y12 + z0 z1 + ε(z1 z2 − x1 x2 )
x21 + y1 y2 + z12 + ε(y0 y1 + z0 z1 )
x1 x2 + y22 + z1 z2 + ε(y12 + z12 )
x1 x2 + y02 + z02 − z1 z2
x22 + y0 y1 + z0 z1 − z22
For ε 6= 0 we can write the ideal as
x20 + y0 y1 + z02 + ε(z12 − x21 )
x0 x1 + y12 + z0 z1 + ε(z1 z2 − x1 x2 )
x21 + y1 y2 + z12 + ε(z22 − x22 )
x0 x1 + y12 + z0 z1 + ε(y02 + z02 )
x21 + y1 y2 + z12 + ε(y0 y1 + z0 z1 )
x1 x2 + y22 + z1 z2 + ε(y12 + z12 )
We can describe this deformation in the following way. Write Q = sQs + tQt .
The two times three equations above are obtained by rolling factors from sP −
εQt and tP + εQs . We may generalize this example.
Lemma 2.4. Let V be a complete intersection of divisors of type aH − b1 R,
aH − b2 R, given by equations P , Q. If it is possible to write Q = sQs +
tb1 −b2 −1 Qt then the equations sP − εQt , tb1 −b2 −1 P + εQs give a deformation
to a complete intersection of type aH − (b1 − 1)R, aH − (b2 + 1)R.
In general one has to combine such a deformation with a deformation of the
scroll.
Documenta Mathematica 7 (2002) 185–226

198

Jan Stevens

(2.5) Non-scrollar deformations.
Example 2.6. As mentioned before such deformations must exist in weight
zero for trigonal cones. We proceed with the explicit computation of embedded
deformations. We start from the normal bundle exact sequence
0 −→ NS/C −→ NC −→ NS ⊗ OC −→ 0 .
As C is a curve of type 3H − (g − 4)R on S we have that C · C = 3g + 6
and H 1 (C, NS/C ) = 0. So we are interested in H 0 (C, NS ⊗ OC ), and more
particularly in the cokernel of the map H 0 (S, NS ) −→ H 0 (C, NS ⊗ OC ), as
H 0 (S, NS ) gives deformations of the scroll.
Proposition 2.7. The cokernel of the map H 0 (S, NS ) −→ H 0 (C, NS ⊗ OC )
has dimension g − 4.
Proof . An element of H 0 (C, NS ⊗ OC ) is a function ϕ on the equations of the
scroll such that the generators of the module of relations map to zero in OC
and it lies in the image of H 0 (S, NS ) if the function values can be lifted to OS
such that the relations map to 0 ∈ OS . Therefore we perform our computations
in OS .
We have to introduce some more notation. Using the equations described
in (1.3) we have three types of scrollar equations, fi,j = xi xj+1 − xi+1 xj ,
gi,j = yi yj+1 − yi+1 yj and mixed equations hi,j = xi yj+1 − xi+1 yj . The scrollar
relations come from doubling a row in the matrix and there are two ways to
do this. The equations resulting from doubling the top row can be divided by
s, and the other ones by t, so the result is the same.
A relation involving only equations of type fi,j gives the condition
xse1 −i−1 ti ϕ(fj,k ) − xse1 −j−1 tj ϕ(fi,k ) + xse1 −k−1 tk ϕ(fi,j ) = 0 ∈ OC
which may be divided by x. As the image ϕ(fi,j ) is quadratic in x and y the
resulting left hand side cannot be a multiple of the equation of C, so we have
se1 −i−1 ti ϕ(fj,k ) − se−1−j−1 tj ϕ(fi,k ) + se1 −k−1 tk ϕ(fi,j ) = 0 ∈ OS
and the analogous equation involving only the gi,j equations.
For the mixed equations we get
xse1 −i−1 ti ϕ(hj,k ) − xse1 −j−1 tj ϕ(hi,k ) + yse2 −k−1 tk ϕ(fi,j ) = ψi,j;k P ∈ OS
with ψi,j;k of degree e1 + e2 − 3 = g − 5 and analogous ones involving gi,j with
coefficients ψi;j,k . These coefficients are not independent, but satisfy a systems
of equations coming from the syzygies between the relations. They can also be
verified directly. We obtain
se1 −i−1 ti ψj,k;l − se1 −j−1 tj ψi,k;l + se1 −k−1 tk ψi,j;l = 0 ∈ OS
Documenta Mathematica 7 (2002) 185–226

199

Rolling Factors Deformations
and

xse1 −i−1 ti ψj;k,l − xse1 −j−1 tj ψi;k,l + yse2 −k−1 tk ψi,j;l − yse2 −l−1 tl ψi,j;k = 0
The last set of equations shows that se2 −k−1 tk ψi,j;l = se2 −l−1 tl ψi,j;k (rolling
factors!) and therefore ψi,j;k = se2 −k−1 tk ψi,j; with ψi,j; of degree e1 − 2. This
yields the equations
se1 −i−1 ti ψj,k; − se1 −j−1 tj ψi,k; + se1 −k−1 tk ψi,j; = 0
Our next goal is to express all ψi,j; in terms of the ψi,i+1; (where 0 ≤ i ≤
e1 − 2). First we observe by using the last equation for the triples (0, i, i + 1)
and (i, i + 1, e1 − 1) that ψi,i+1; is divisible by ti and by se1 −i−2 so ψi,i+1; =
se1 −i−2 ti ci for some constant ci . By induction it then follows that ψi,j; =
se1 −i−2 ti cj−1 + se1 −i−3 ti+1 cj−2 + · · · + se1 −j−1 tj−1 ci , so the solution of the
equations depends on e1 − 1 constants. Similarly one finds e2 − 1 constants di
for the ψi;j,k so altogether e1 + e2 − 2 = g − 4 constants.
Finally we can solve for the perturbations of the equations. We give the formulas in the case that all di and all ci but one are zero, say cγ = 1. This implies
that ψi,j; = 0 if γ ∈
/ [i, j) and ψi,j; = se1 −i−j+γ−1 ti+j−γ−1 if γ ∈ [i, j); under
the last assumption ψi,j;k = sg−4−i−j−k+γ ti+j+k−γ−1 . We take ϕ(fi,j ) = 0 if
γ ∈
/ [i, j). It follows that for a fixed k the ϕ(hi,k ) with i ≤ γ are related by
rolling factors, as are the ϕ(hi,k ) with i > γ. This reduces the mixed equations
with fixed k to one, which can be solved for in a uniform way for all k. To this
end we write the equation P as
γ+1 −
(s2e1 −e2 −γ+2 A+
A2e1 −e2 −γ+1 )x3 + y(Be1 +2 x2 + Ce2 +2 xy + D2e2 −e1 +2 y 2 )
γ +t

which we will abbreviate as (s2e1 −e2 −γ+2 A+ + tγ+1 A− )x3 + yE. We set
ϕ(fi,j )
ϕ(fi,j )

= 0,
= se1 −i−j−1+γ ti+j−γ−1 E,

ϕ(gi,j ) = 0,
ϕ(hi,k ) = −se2 −1−k−i+γ ti+k A− x2 ,
ϕ(hi,k ) = s

2e1 +1−i−k i+k−γ−1

t

if γ ∈
/ [i, j)
if γ ∈ [i, j)
if i ≤ γ

+ 2

A x , if i > γ

This is well defined, because all exponents of s and t are positive.

¤
1

A similar computation can be used to show that all elements of T (ν) with
ν > 0 can be written rolling factors type. However, even more is true, they can
be represented as pure rolling factors deformations, see [Drewes–Stevens 1996],
where a direct argument is given.
We generalize the above discussion
divisors of type aH − bi R (with the
µ (1)
z0
. . . zd(1)
1 −1
(1)
z1
. . . zd(1)
1

to the case of a complete intersection of
same a ≥ 2) on a scroll
¶
. . . z0(k) . . . zd(k)
k −1
.
. . . z1(k) . . . zd(k)
k

Documenta Mathematica 7 (2002) 185–226

200

Jan Stevens
(αβ)

(β)
(α)
We have equations fij = zi(α) zj+1
− zi+1
zj(β) . The lowest degree in which non
rolling factors deformations can occur is a − 3. We get the conditions

¡ (βγ) ¢
¡ (αγ) ¢
¡ (αβ) ¢
z (α) sdα −i−1 ti ϕ fjk
− z (β) sdβ −j−1 tj ϕ fik
+ z (γ) sdγ −k−1 tk ϕ fij
X (αβγ)
=
ψijk;l P (l)
l

(αβγ)

with the ψijk;l homogeneous polynomials in (s:t) of degree bl −1. The relations
between these polynomials come from the syzygies of the scroll: we add four of
these relations, multiplied with a term linear in the z (α) ; then the left hand side
becomes zero, leading to a relation (in OS ) between the P (l) . As we are dealing
with a complete intersection, the relations are generated by Koszul relations.
Because the coefficients of the relation obtained are linear in the z (α) , they
cannot lie in the ideal generated by the P (l) (as a ≥ 2), so they vanish and we
obtain for each l equations
(βγδ)

(αγδ)

z (α) sdα −i−1 ti ψjkm;l − z (β) sdβ −j−1 tj ψikm;l
(αβδ)

(αβγ)

+ z (γ) sdγ −k−1 tk ψijm;l − z (δ) sdδ −m−1 tm ψijk;l = 0 .
Here some of the α, . . . , δ may coincide. If e.g. δ is different from α, β and γ,
(αβγ)
then ψijk;l = 0. If there are at least four different indices (e.g. if the scroll
is nonsingular of dimension at least four) then δ can always be chosen in this
way, so all coefficients vanish and every deformation of degree a − 3 is of rolling
factors type.
Suppose now the scroll is a cone over a nonsingular 3-dimensional scroll, i.e.
(αβγ)
we have three different indices at our disposal. Then every ψijk;l with at most
two different upper indices vanishes, and the ones with three different indices
satisfy rolling factors equations. We conclude that for pairwise different α, β, γ
(αβγ)

ψijk;l = sd−i−j−k−3 ti+j+k ψl′
with d = dα + dβ + dγ the degree of the scroll.
Finally, for the cone over a 2-dimensional scroll we get similar computations as
in the trigonal example above.
Proposition 2.8. A tetragonal cone (with g > 5) has non-scrollar deformations of degree −1 if and only if b2 = 0. If the canonical curve lies on a
Del Pezzo surface then the dimension is 1. If the curve is bielliptic then the
dimension is b1 = g − 5.
Proof . First suppose e3 > 0. ThenP
the only possibly non
Pzero coefficients are
the ψl′ , which have degree bl + 2 − ei . As b1 + b2 =
ei − 2 they do not
vanish iff b2 = 0. In this case the computation yields one non rolling factors
deformation of the Del Pezzo surface on which the curve lies.
Documenta Mathematica 7 (2002) 185–226

201

Rolling Factors Deformations

If e3 = 0, then b2 = 0. For a bielliptic curve the methods above yield (e1 −
1) + (e2 − 1) = b1 = g − 5 non-scrollar deformations (a detailed computation is
given in [Brawner 1996]). Suppose now that the curve lies on a (singular) Del
Pezzo surface. If b1 = e1 > e2 = 2 then the equation P contains the monomial
xz with nonzero coefficient, which we take to be 1, while there is no monomial
yz. After a coordinate transformation we may assume that the same holds in
case e1 = e2 = 2. Let ϕ(hi,k ) ≡ ζi,k z mod (x, y). In the equation
xse1 −i−1 ti ϕ(gj,k ) − yse2 −j−1 tj ϕ(hi,k ) + yse2 −k−1 tk ϕ(hi,j ) = ψi;j,k P
holding in OS the monomial yz occurs only on the left hand side, which shows
that the ζi,k are of rolling factors type in the first index. Being constants, they
vanish. This means that in the equation
xse1 −i−1 ti ϕ(hj,k ) − xse1 −j−1 tj ϕ(hi,k ) + yse2 −k−1 tk ϕ(fi,j ) = ψi,j;k P
the monomial xz does not occur on the left hand side and therefore ψi,j;k = 0.
We find only e2 − 1 = 1 non rolling factors deformation. If e1 = 2, e2 = 1 we
find one deformation. Finally, if e1 = 3, e2 = 1 then there is only one type of
mixed equation. We have two constants c0 and c1 . Let the coefficient of xz in
P be p0 s + p1 t. We obtain the equations
sζ1,0 − tζ0,0 = c0 (p0 s + p1 t)
sζ2,0 − tζ1,0 = c1 (p0 s + p1 t)
from which we conclude that p0 c0 + p1 c1 = 0, giving again only one non rolling
factors deformation.
¤
(2.9) Rolling factors deformations of degree −1.
We look at the miniversal deformation of the scroll:
µ
z0(1)
...
zd(1)
zd(1)
z0(2) . . .
zd(k)
1 −2
1 −1
k −2
(2)
(k)
(1)
(1)
(1)
(1)
(1)
z1
. . . zdk −1 + ζd(k)
z 1 + ζ1
. . . zd1 −1 + ζd1 −1
zd1
k −1

zd(k)
k −1
zd(k)
k

¶

To compute which of those deformations can be lifted to deformations of a
complete intersection on the scroll we have to compute perturbations of the
additional equations.
We assume that we have a complete intersection of divisors of type aH − bi R
(with the same a ≥ 2).
Extending the notation introduced before we write the columns in the matrix
symbolically as (zα , zα+1 +ζα+1 ). In order that this makes sense for all columns
we introduce dummy variables ζ0(i) and ζd(i)i with the value 0.
The Koszul type relations give no new conditions, but the relation
Pm+1 zβ − Pm zβ+1 −

X

pα,m fβα = 0

α

Documenta Mathematica 7 (2002) 185–226

202

Jan Stevens

′
of Pm :
gives as equation in the local ring for the perturbations Pm
′
′
Pm+1
zβ − P m
zβ+1 −

X

pα,m (ζα+1 zβ − zα ζβ+1 ) = 0 .

α

In particular
we see that we can look at one equation on the scroll at a time.
P
As
pα,m zα = Pm the coefficient of ζβ+1 vanishes. Because tzβ − szβ+1 = 0
we get a condition which is independent of β:
′
′
sPm+1
− tPm
−s

X

pα,m ζα+1 = 0

α

This has to hold in the local ring, but as the degree of the pα,m is lower than
that of the equations defining the complete intersection on the scroll (here we
use the assumption that all degrees a are equal), it holds on the scroll. From
it we derive the equation
sb Pb′ − tb P0′ =

b−1 X
X

sm+1 tb−m−1 pα,m ζα+1

(S)

m=0 α

which has to be solved with Pb′ and P0′ polynomials in the zα of degree a − 1.
We determine the monomials on the right hand side.
The result depends on the chosen equations, but only on P0 and Pb and not on
the intermediate ones, provided they are obtained by rolling factors.
Example 2.10. Let b = 4. We take variables yi = s3−i ti y, zi = s3−i ti z with
deformations ηi , ζi , and roll from y0 z0 to y2 z2 in two different ways:
y 0 z0 → y 1 z0 → y 1 z1 → y 2 z1 → y 2 z2
y 0 z0 → y 0 z1 → y 0 z2 → y 1 z2 → y 2 z2
This gives as right-hand side of the equation (S) in the two cases
s4 t3 zη1 + s4 t3 yζ1 + s5 t2 zη2 + s5 t2 yζ2
s4 t3 yζ1 + s5 t2 yζ2 + s4 t3 zη1 + s5 t2 zη2
which is the same expression. Similarly, if we roll from z02 to z22 we get
2s4 t3 zζ1 + 2s5 t2 zζ2
However, if we roll in the last step from y1 z2 to y1 z3 we get
s4 t3 yζ1 + s5 t2 yζ2 + s4 t3 zη1
(remember that we have no deformation parameter ζ3 ).
Documenta Mathematica 7 (2002) 185–226

Rolling Factors Deformations

203

To analyze the general situation it is convenient to use multi-index notation.
The equation P of a divisor in |aH − bR| may then be written as
P =

X

|I|=a

he,Ii−b

X

pI,j she,Ii−b−j tj z I .

j=0

Here e = (e1 , . . . , ek ) is the vector of degrees and z I stands for (z (1) )i1 · . . . ·
(z (k) )ik .
Proposition 2.11. The lifting condition for the equations Pm is that for each
I with |I| = a − 1 and he, Ii < b − 1 the following b − he, Ii − 1 linear equations
hold:
k he,I+δ
X
Xl i−b
(l)
=0,
(il + 1)pI+δl ,j ζj+n
l=1

j=0

where 0 < n < b − he, Ii
′

′

Proof . We look at a monomial she,I i−b−j tj z I . In rolling from P0 to Pm
we go from zA to zA+B . Here we write a monomial as product of a factors:
zα1 · · · zαa with i′l factors of type l. Let I ′ = I + δl with δl the lth unit vector.
The monomial leads to an expression in which the coefficient of z I is
X

βq
X

she,Ii+r−j+αq tb−r+j−αq ζαq +r

{q|αq of type l} r=1

We stress that the choice of αq can be very different for different j.
We collect all contributions and look at the coefficient of she,Ii+n tb−n z I with
0 < n < b − he, Ii. This cannot be realized as left-hand side of equation (S).
Because b − n = b − r + j − αq this coefficient is
k he,I+δ
X
Xl i−b
l=1

(l)
=0,
(il + 1)pI+δl ,j ζj+n

j=0

We note that all terms really occur: in rolling from zA to zA+B we have to
increase the qth factor sufficiently many times, because he, Ii < b − 1.
¤
Example 2.12: trigonal cones.
equation

Let the curve be given by the bihomogeneous

F = A2a−m+2 z 3 + Ba+2 z 2 w + Cm+2 zw2 + D2m−a+2 w3
then there are only conditions for I = (0, 2), i.e. for w 2 , as a+m = g −2 > b−1.
So if 2m < b − 1 = g − 5 we get b − 2m − 1 = a − m − 3 equations on the
deformation variables ζ1 , . . . , ζa−1 , ω1 , . . . , ωm−1
m+2
X
j=0

cj ζj+n + 3

2m−a+2
X

dj ωj+n = 0 ,

j=0

Documenta Mathematica 7 (2002) 185–226

204

Jan Stevens

as stated in [Drewes–Stevens 1996, 3.11]. We have a system of linear equations
so we can write the coefficient matrix. It consists of two blocks ( C | D ) with C
of the form
c

0

0

0




0

0
0

c1
c0
0

c2
c1
c0

...
...
...
..
.

cm+2
cm+1
cm

0
cm+2
cm+1

0
0
0

0
0
0

...
...
...

c0
0
0

c1
c0
0

0
0
cm+2

...
...
...
..
.

0
0
0

0
0
0

0
0
0

c2
c1
c0

...
...
...

cm+2
cm+1
cm

0
cm+2
cm+1

0
0
cm+2












and D similarly. Obviously this system has maximal rank.
The Proposition gives a system of linear equations and we call the coefficient
matrix lifting matrix. It was introduced for tetragonal cones in [Brawner 1996].
In general the lifting matrix will have maximal rank, but it is a difficult question
to decide when this happens.
Example 2.13: trigonal K3s. We take the invariants (e, e, e), b = 3e − 2 with
e ≥ 3. The K3 lies on P2 × P1 and is given by an equation of bidegree (3, 2).
Now there are six Is with |I| = 2 each giving rise to e − 3 equations in 3(e − 1)
deformation variables. In general the matrix has maximal rank, but for special
surfaces the rank can drop. Consider an equation of type p1 x3 +p2 y 3 +p3 z 3 with
the pi quadratic polynomials in (s:t) without common or multiple zeroes. Then
the surface is smooth. The lifting equations corresponding to the quadratic
monomials xy, xz and yz vanish identically and the lifting matrix reduces to
a block-diagonal matrix of rank 3(e − 3). The kernel has dimension 6, but the
corresponding deformations are obstructed: an extension of the K3 would be
a Fano 3-fold with isolated singularities lying as divisor of type 3H − bR on
a scroll S(e
P 1 , e2 , e3 , e4 ), and a computation reveals that such a Fano can only
exist for
ei ≤ 8.
We can say something more for the lifting conditions coming from one quadratic
equation.

Proposition 2.14. The lifting matrix for one quadratic equation has dependent rows if and only if the generic fibre has a singular point on the subscroll
Bb−1 .
Proof .
The equation P on the scroll can be written in the form t zΠz
with Π a symmetric k × k matrix with polynomials in (s : t) as entries.
The condition that there is a singular section of the form z = (0, . . . , 0,
z (l+1) (s, t), . . . , z (k) (s, t)) with el > b − 1 is that t zΠ = 0 or t z>l Π>l = 0 where
z>l = (z (l+1) (s, t), . . . , z (k) (s, t)) and Π>l is the matrix consisting of the last
k − l rows of Π. The resulting system of equations for the coefficients of the
polynomials z (i) (s, t) gives exactly the lifting matrix.
¤
Documenta Mathematica 7 (2002